arXiv subjects
Publications and source records attributed to Michael Wiemeler.
We prove conjectures due to Fukaya--Yamaguchi and Kapovitch--Petrunin--Tuschmann for fundamental groups of (almost) non-negatively curved closed manifolds under symmetry assumptions.
We classify Riemannian foliations of closed manifolds homotopy equivalent to CROSSes.
We prove vanishing results for Witten genera of string generalized complete intersections in homogeneous $\text{Spin}^c$-manifolds and in other $\text{Spin}^c$-manifolds with Lie group actions. By applying these results to Fano manifolds with second Betti number equal to one we get new evidence for a conjecture of Stolz.
We identify a link between regular matroids and torus representations all of whose isotropy groups have an odd number of components. Applying Seymour's 1980 classification of the former objects, we obtain a classification of the latter. In addition, we prove optimal upper bounds for the cogirth of regular matroids up to rank nine, and we apply this to prove the existence of fixed-point sets of circles with large dimension in a torus representation with this property up to rank nine. Finally, we apply these results to prove new obstructions to the existence of Riemannian metrics with positive sectional curvature and torus symmetry.
We show that closed, simply connected, positively curved 10-manifolds with effective, isometric actions of $3$-dimensional tori are homotopy spheres or homotopy complex projective spaces.
We discuss the rigidity of elliptic genera for non-spin manifolds $M$ with $S^1$-action. We show that if the universal covering of $M$ is spin, then the universal elliptic genus of $M$ is rigid. Moreover, we show that there is no condition which only depends on $π_2(M)$ that guarantees the rigidity in the case that the universal covering of $M$ is non-spin.
In 1996 Stolz conjectured that a string manifold with positive Ricci curvature has vanishing Witten genus. Here we prove this conjecture for toric string Fano manifolds and for string torus manifolds admitting invariant metrics of non-negative sectional curvature.
We study smooth, closed orientable $S^1$-manifolds $M$ with exactly $3$ fixed points. We show that the dimension of $M$ is of the form $4\cdot 2^a$ or $8\cdot(2^a+2^b)$ with $a,b\geq 0$ and $a\neq b$. Moreover, under the extra assumption that $M$ is spin or unitary we determine exactly the dimension of $M$.
Let $M^d$ be a simply connected spin manifold of dimension $d \geq 5$ admitting Riemannian metrics of positive scalar curvature. Denote by $\mathcal{R}^+(M^d)$ the space of such metrics on $M^d$. We show that $\mathcal{R}^+(M^d)$ is homotopy equivalent to $\mathcal{R}^+(S^d)$, where $S^d$ denotes the $d$-dimensional sphere with standard smooth structure. We also show a similar result for simply connected non-spin manifolds $M^d$ with $d\geq 5$ and $d\neq 8$. In this case let $W^d$ be the total space of the non-trivial $S^{d-2}$-bundle with structure group $SO(d-1)$ over $S^2$. Then $\mathcal{R}^+(M^d)$ is homotopy equivalent to $\mathcal{R}^+(W^d)$.
We study spaces and moduli spaces of Riemannian metrics with non-negative Ricci or non-negative sectional curvature on closed and open manifolds. We construct, in particular, the first classes of manifolds for which these moduli spaces have non-trivial rational homotopy, homology and cohomology groups. We also show that in every dimension at least seven (respectively, at least eight) there exist closed (respectively, open) manifolds for which the space and moduli space of Riemannian metrics with non-negative sectional curvature has infinitely many path components. A completely analogous statement holds for spaces and moduli spaces of non-negative Ricci curvature metrics.
We study locally standard $T^k$-manifolds $M$. In particular, we study the case where there is a continuous section to the orbit map $π: M \rightarrow M/T$. We give a classification of $T^k$-manifolds satisfying these conditions up to equivariant diffeomorphism.
In this paper we study non-negatively curved and rationally elliptic GKM$_4$ manifolds and orbifolds. We show that their rational cohomology rings are isomorphic to the rational cohomology of certain model orbifolds. These models are quotients of isometric actions of finite groups on non-negatively curved torus orbifolds. Moreover, we give a simplified proof of a characterisation of products of simplices among orbit spaces of locally standard torus manifolds. This characterisation was originally proved in [Wiemeler, Torus manifolds and non-negative curvature, arXiv:1401.0403] and was used there to obtain a classification of non-negatively curved torus manifolds.
A 1930s conjecture of Hopf states that an even-dimensional compact Riemannian manifold with positive sectional curvature has positive Euler characteristic. We prove this conjecture under the additional assumption that the isometry group has rank at least five. The fundamental new tool used to achieve this is a reduction to, and structural results concerning, a representation theoretic problem involving torus representations all of whose isotropy groups are connected.
Let $M$ be a simply connected spin manifold of dimension at least six which admits a metric of positive scalar curvature. We show that the observer moduli space of positive scalar curvature metrics on $M$ has non-trivial higher homotopy groups. Moreover, denote by $\mathcal{M}_0^+(M)$ the moduli space of positive scalar cuvature metrics on $M$ associated to the group of orientation-preserving diffeomorphisms of $M$. We show that if $M$ belongs to a certain class of manifolds which includes $(2n-2)$-connected $(4n-2)$-dimensional manifolds, then the fundamental group of $\mathcal{M}_0^+(M)$ is non-trivial.
In this note we classify simply connected rationally elliptic compact toric orbifolds up to algebraic isomorphism.
We construct geometric generators of the effective $S^1$-equivariant Spin- (and oriented) bordism groups with two inverted. We apply this construction to the question of which $S^1$-manifolds admit invariant metrics of positive scalar curvature. It turns out that, up to taking connected sums with several copies of the same manifold, the only obstruction to the existence of such a metric is an $\hat{A}$-genus of orbit spaces. This $\hat{A}$-genus generalizes a previous definition of Lott for orbit spaces of semi-free $S^1$-actions. As a further application of our results, we give a new proof of the vanishing of the $\hat{A}$-genus of a Spin manifold with non-trivial $S^1$-action originally proven by Atiyah and Hirzebruch. Moreover, based on our computations we can give a bordism-theoretic proof for the rigidity of elliptic genera originally proven by Taubes and Bott--Taubes.
A classical theorem due to Quillen (1969) identifies the unitary bordism ring with the Lazard ring, which classifies the universal one-dimensional commutative formal group law. We prove an equivariant generalization of this result by identifying the homotopy theoretic $\mathbb{Z}/2$-equivariant unitary bordism ring, introduced by tom Dieck (1970), with the $\mathbb{Z}/2$-equivariant Lazard ring, introduced by Cole-Greenlees-Kriz (2000). Our proof combines a computation of the homotopy theoretic $\mathbb{Z}/2$-equivariant unitary bordism ring due to Strickland (2001) with a detailed investigation of the $\mathbb{Z}/2$-equivariant Lazard ring.
We show that the higher homotopy groups of the moduli space of torus-invariant positive scalar curvature metrics on certain quasitoric manifolds are non-trivial.